One of the most powerful ideas in array processing is the duality between discrete-time signal processing and array signal processing: whereas discrete-time systems sample signals in time, arrays sample signals in space. Discrete-time frequency is analogous to wavenumber, which depends on direction of arrival; spatially selective processing, such as beamforming, is analogous to discrete-time filtering; and the Fourier transform can be used to perform spatial analysis. The reader is referred to Corey et al., Spatial Sigma-Delta Signal Acquisition for Wideband Beamforming Arrays, Proc. of the 20th International ITG Workshop, pp. 1-7 (2016) (hereinafter, “Corey 2016”), incorporated herein by reference, for background information.
Error shaping as a discrete-time signal processing method has rarely been applied to array processing The term “error shaping” is used herein, and in any appended claims, instead of the more popular “noise shaping,” to distinguish quantization errors from other noise sources, such as interfering signals and circuit noise. Error shaping may be employed to reduce quantization error in systems with coarse or even single-bit quantization. A delta-sigma analog-to-digital converter (ADC), described by Schreier et al., Understanding Delta-Sigma Data Converters, Wiley (2005) (hereinafter, “Schreier 2005”) and by Geerts et al., Design of Multi-bit Delta-Sigma A/D Converters, Springer (2006) (hereinafter, “Geerts 2006”), is considered with reference to FIG. 1A, where it is designated generally by numeral 11. Both of the foregoing books are incorporated herein by reference.
Delta-sigma analog-to-digital converter 11 uses analog feedback to shape quantization errors to higher frequencies. Referring to FIG. 1B, if the input signal in a signal band 13 is bandlimited in frequency and is oversampled in time, then most of the energy in the shaped error signal 15 is concentrated at high frequencies that do not contain input signal information. The output of the error-shaping modulator can be lowpass filtered to remove most of the quantization error and then downsampled to the Nyquist rate.
The same principle can be applied to array design: by oversampling in space and propagating error signals from one sensor to the next, an array can use low-precision quantizers to produce a high-precision digital output. As a motivating example for spatial error shaping, FIG. 2 shows a beamforming array designated generally by numeral 21 that is the spatial equivalent of the first-order sigma-delta ADC in FIG. 1. Quantization errors propagate from one array channel to the next and are thereby shaped to higher wavenumbers. If the array is oversampled in space, that is, if the element spacing is smaller than one half wavelength, then the quantization errors will be shaped outside the wavenumber range that propagating waves can occupy. In this example, a beamformer plays the role of the lowpass filter, removing high-wavenumber components.
Because they use fewer comparator circuits, error-shaping ADCs require much less area and power than architectures with high-resolution quantizers and are widely used in low-speed applications in which oversampling is feasible. While oversampling and error-shaping ADCs are well studied and spatial error shaping is widely used in image processing techniques such as Floyd-Steinberg dithering, there has been comparatively little work on spatial oversampling and error shaping in the array processing literature. It was reported by Halsig et al., Spatial oversampling in LOS MIMO systems with 1-bit quantization at the receiver, Int. ITG Conf. on Systems, Communication and Coding, pp. 1-6 (2017), incorporated herein by reference, that, even without error shaping, spatial oversampling can compensate for quantization errors in coarsely quantized communication receivers. In Yeang et al., Dense transmit and receive phased arrays, Proc. of the IEEE Int. Symp. on Phased Arrays pp. 934-39, (2004) and related papers, spatial error shaping is used to quantize phase shift values in a narrowband phased array. In Scholnik et al., Spacio-temporal delta-sigma modulation for shared wideband transmit arrays, Proc. IEEE Radar Conf., pp. 85-90 (2004), for example, the authors apply digital space-time error shaping to the signals fed into a transmit array with coarse digital-to-analog converters.
The idea of applying error shaping directly to the received signals in a sensor array was proposed independently in Corey 2016 and in Barac et al., Spatial sigma-delta modulation in a massive MIMO cellular system, Master's thesis, Chalmers University of Technology and the University of Gothenburg (2016) (hereinafter, “Barac 2016”), incorporated herein by reference. Corey 2016 analyzed the performance of space-time error shaping for wideband signals, such as audio, while Barac 2016 focused on narrowband multiple-input multiple-output communication systems. Both works were restricted to signals sampled at the Nyquist rate, without mixing, and to simple delay-and-sum beamformers.